Complex number arithmetic

If the macro constant __STDC_NO_COMPLEX__ is defined by the implementation, the complex types, the header <complex.h> and all of the names listed here are not provided.

(since C11)

The C programming language, as of C99, supports complex number math with the three built-in types double _Complex, float _Complex, and long double _Complex (see _Complex). When the header <complex.h> is included, the three complex number types are also accessible as double complex, float complex, long double complex.

In addition to the complex types, the three imaginary types may be supported: double _Imaginary, float _Imaginary, and long double _Imaginary (see _Imaginary). When the header <complex.h> is included, the three imaginary types are also accessible as double imaginary, float imaginary, and long double imaginary.

Standard arithmetic operators +, -, *, / can be used with real, complex, and imaginary types in any combination.

A compiler that defines `__STDC_IEC_559_COMPLEX__` is recommended, but not required to support imaginary numbers. POSIX recommends checking if the macro _Imaginary_I is defined to identify imaginary number support.	(since C99) (until C11)
Imaginary numbers are supported if `__STDC_IEC_559_COMPLEX__` or `__STDC_IEC_60559_COMPLEX__`(since C23) is defined.	(since C11)

Defined in header <complex.h>

The following function names are potentially(since C23) reserved for future addition to complex.h and are not available for use in the programs that include that header: cerf, cerfc, cexp2, cexpm1, clog10, clog1p, clog2, clgamma, ctgamma, csinpi, ccospi, ctanpi, casinpi, cacospi, catanpi, ccompoundn, cpown, cpowr, crootn, crsqrt, cexp10m1, cexp10, cexp2m1, clog10p1, clog2p1, clogp1(since C23), along with their -f and -l suffixed variants.

Although the C standard names the inverse hyperbolics with "complex arc hyperbolic sine" etc., the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct names are "complex inverse hyperbolic sine" etc. Some authors use "complex area hyperbolic sine" etc.

A complex or imaginary number is infinite if one of its parts is infinite, even if the other part is NaN.

A complex or imaginary number is finite if both parts are neither infinities nor NaNs.

A complex or imaginary number is a zero if both parts are positive or negative zeroes.

While MSVC does provide a <complex.h> header, it does not implement complex numbers as native types, but as structs, which are incompatible with standard C complex types and do not support the +, -, *, / operators.

Example

Run this code

#include <complex.h>
#include <stdio.h>
#include <tgmath.h>
 
int main(void)
{
    double complex z1 = I * I;     // imaginary unit squared
    printf("I * I = %.1f%+.1fi\n", creal(z1), cimag(z1));
 
    double complex z2 = pow(I, 2); // imaginary unit squared
    printf("pow(I, 2) = %.1f%+.1fi\n", creal(z2), cimag(z2));
 
    double PI = acos(-1);
    double complex z3 = exp(I * PI); // Euler's formula
    printf("exp(I*PI) = %.1f%+.1fi\n", creal(z3), cimag(z3));
 
    double complex z4 = 1 + 2 * I, z5 = 1 - 2 * I; // conjugates
    printf("(1+2i)*(1-2i) = %.1f%+.1fi\n", creal(z4 * z5), cimag(z4 * z5));
}

Output:

I * I = -1.0+0.0i
pow(I, 2) = -1.0+0.0i
exp(I*PI) = -1.0+0.0i
(1+2i)*(1-2i) = 5.0+0.0i

References

C23 standard (ISO/IEC 9899:2024):

6.10.8.3/1/2 __STDC_NO_COMPLEX__ (p: TBD)

6.10.8.3/1/2 __STDC_IEC_559_COMPLEX__ (p: TBD)

7.3 Complex arithmetic <complex.h> (p: TBD)

7.25 Type-generic math <tgmath.h> (p: TBD)

7.31.1 Complex arithmetic <complex.h> (p: TBD)

Annex G (normative) IEC 60559-compatible complex arithmetic (p: TBD)

C17 standard (ISO/IEC 9899:2018):

6.10.8.3/1/2 __STDC_NO_COMPLEX__ (p: 128)

6.10.8.3/1/2 __STDC_IEC_559_COMPLEX__ (p: 128)

7.3 Complex arithmetic <complex.h> (p: 136-144)

7.25 Type-generic math <tgmath.h> (p: 272-273)

7.31.1 Complex arithmetic <complex.h> (p: 391)

Annex G (normative) IEC 60559-compatible complex arithmetic (p: 469-479)

C11 standard (ISO/IEC 9899:2011):

6.10.8.3/1/2 __STDC_NO_COMPLEX__ (p: 177)

6.10.8.3/1/2 __STDC_IEC_559_COMPLEX__ (p: 177)

7.3 Complex arithmetic <complex.h> (p: 188-199)

7.25 Type-generic math <tgmath.h> (p: 373-375)

7.31.1 Complex arithmetic <complex.h> (p: 455)

Annex G (normative) IEC 60559-compatible complex arithmetic (p: 532-545)

C99 standard (ISO/IEC 9899:1999):

6.10.8/2 __STDC_IEC_559_COMPLEX__ (p: 161)

7.3 Complex arithmetic <complex.h> (p: 170-180)

7.22 Type-generic math <tgmath.h> (p: 335-337)

7.26.1 Complex arithmetic <complex.h> (p: 401)

Annex G (informative) IEC 60559-compatible complex arithmetic (p: 467-480)

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Complex number arithmetic

Contents

Types

The imaginary constant

Manipulation

Exponential functions

Power functions

Trigonometric functions

Hyperbolic functions

Notes

Example

References

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